Query[LeviDecomposition] - check that a pair of subalgebras define a Levi decomposition of a Lie algebra
Query([R, S], "LeviDecomposition")
R - a list of independent vectors in a Lie algebra 𝔤
S - a list of independent vectors in a Lie algebra 𝔤
A pair of subalgebras R, S in a Lie algebra 𝔤 define a Levi decomposition if R is the radical of 𝔤, S is a semi-simple subalgebra, and g = R ⊕ S (vector space direct sum). Since the radical is an ideal we have R,.R ⊂R, R, S ⊂R, and S, S ⊂S. The radical R is unique, the semisimple subalgebra S in a Levi decomposition is not.
Query([R, S], "LeviDecomposition") returns true if the pair R, S is a Levi decomposition of 𝔤 and false otherwise.
The command Query is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).
We initialize three different Lie algebras and print their multiplication tables.
L1 ≔ _DG⁡LieAlgebra,Alg1,3,1,3,1,1,2,3,1,1,2,3,2,1:
L2 ≔ _DG⁡LieAlgebra,Alg2,3,1,2,1,1,1,3,2,−2,2,3,3,1:
L3 ≔ _DG⁡LieAlgebra,Alg3,5,1,3,2,−1,1,4,1,−1,2,4,2,1,2,5,1,−1,3,4,3,2,3,5,4,−1,4,5,5,2:
Alg1 is solvable and therefore the radical is the entire algebra.
R1 ≔ x1,x2,x3:S1 ≔ :
Alg2 is semisimple and therefore the radical is the zero subalgebra.
R2 ≔ :S2 ≔ y1,y2,y3:
Alg3 has a non-trivial Levi decomposition.
R3 ≔ z1,z2:S3 ≔ z3,z4,z5:
It is easy to see that in this last example the Levi decomposition is not unique. First we find the general complement to the radical R3 using the ComplementaryBasis program.
SS0 ≔ ComplementaryBasis⁡R3,z1,z2,z3,z4,z5,k
Next we determine for which values of the parameters k1, k2, k3, k4, k5, k6 for which he subspace SS0 is a Lie subalgebra. We find that k1=0, k2= k3, k4 = −k5, k6 =0.
TF,Eq,SOL,SubAlgList ≔ Query⁡SS0,Subalgebra
S4 ≔ SubAlgList1
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